Parameter identification for the simulation of the periodontal ligament during the initial phase of orthodontic tooth movement

Abstract

The paper is concerned with simulation of the periodontal ligament response to force in the initial phase of orthodontic tooth movement. This is based on two previous investigations, a in vitro experiment with specimens of porcine mandibular premolars and a in vivo experiment on human upper first incisors. For the curve fit of the in vitro experiment a model function, assuming viscoelasticity, was introduced. The viscoelastic model function was augmented by a ramp rise time term, to account for observed dependence of the response on actuator velocity, and a previous load history term, to account for the effect of the previous tests on the current test. The correlation coefficient of a curve fit for all tests grouped together was $R^2=0.98.$ Next, a curve fit of the in vivo experiment was done. Good correlation was found for a simplified model function, without viscoelastic term ($R^2=0.96$). For both tests, in vitro and in vivo, the ramp rise time term improved correlation. A finite element model of the specimen of the in vitro experiment was created. For the PDL a hyperelastic constitutive model for compressible material was used and model parameters were identified. The present work indicates that the macroscopic response of the periodontal ligament to an external load can be simulated with a poro-visco-hyperelastic model. The simulation showed that poroelastic behaviour will gradually cease when viscoelastic relaxation progresses. This followed also from dimensionless analysis. As a consequence, for slow loading, or if initial response to fast loading is not of interest, a visco-hyperelastic model may suffice. To identify parameters of the finite element model several optimisation problems were solved. A model function, which can be regarded as a reduced order model, allowed a full factorial experiment (analysis) at low cost, to identify initial parameters. The thus found parameters were further refined with an optimum interpolation meta-model. That is, for limited number of parameter combinations the response was simulated with the finite element model and a refined parameter study was conducted by means of optimal interpolation. The thus found optimal parameters were verified by simulation with the finite element model. Optimal interpolation is computationally cheap, which allowed full factorial experiments at low cost.

Keywords:

• Periodontal ligament (PDL)
• initial phase of orthodontic tooth movement (OTM)
• test and simulation
• poro-visco-hyperelastic model
• optimisation
• parameter identification
• reduced order model

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 $\frac{\partial T(\lambda )}{\partial \lambda }=E+\beta T(\lambda )$ with T traction, λ stretch, E Young’s modulus, β a constant.

2 In the paper of Bergomi et al. (2009) the $n/n_0$ ratio is stated without exponent. However, the original paper of Argoubi and Shirazi-Adl (1996) and the review paper of Riches et al. (2002) show the exponent two. The quadratic term was used here.